In applications of AI systems where exact definitions of the important notions of the application domain are hard to come by, the use of traditional logic-based knowledge representation languages such as Description Logics may lead to very large and unintuitive definitions, and high complexity of reasoning. To overcome this problem, we define new concept constructors that allow us to define concepts in an approximate way. To be more precise, we present a family $\tau \mathcal{EL}(m)$ of extensions of the lightweight Description Logic $\mathcal{EL}$ that use threshold constructors for this purpose. To define the semantics of these constructors we employ graded membership functions m, which for each individual in an interpretation and concept yield a number in the interval $[0,1]$ expressing the degree to which the individual belongs to the concept in the interpretation. Threshold concepts $C_{\bowtie t}$ for $\bowtie \in \{<,\le ,>,\ge \}$ then collect all the individuals that belong to C with degree ⋈t. The logic $\tau \mathcal{EL}(m)$ extends $\mathcal{EL}$ with threshold concepts whose semantics is defined relative to a function m. To construct appropriate graded membership functions, we show how concept measures ∼ (which are graded generalizations of subsumption or equivalence between concepts) can be used to define graded membership functions $m_{\sim }$. Then we introduce a large class of concept measures, called simi-d, for which the logics $\tau \mathcal{EL}(m_{\sim })$ have good algorithmic properties. Basically, we show that reasoning in $\tau \mathcal{EL}(m_{\sim })$ is NP/coNP-complete without TBox, PSpace-complete w.r.t. acyclic TBoxes, and ExpTime-complete w.r.t. general TBoxes. The exception is the instance problem, which is already PSpace-complete without TBox w.r.t. combined complexity. While the upper bounds hold for all elements of simi-d, we could prove some of the hardness results only for a subclass of simi-d. This article considerably improves on and generalizes results we have shown in three previous conference papers and it provides detailed proofs of all our results.

在人工智能系统的应用中，很难获得应用领域重要概念的准确定义，使用传统的基于逻辑的知识表示语言（例如描述逻辑）可能会导致定义非常大且不直观，推理的复杂性也很高。为了克服这个问题，我们定义了新的概念构造器，使我们能够以近似的方式定义概念。更准确地说，我们呈现的是一个家庭$\tau \mathcal{EL}（米）$轻量级描述逻辑的扩展$\mathcal{EL}$为此目的使用阈值构造函数。为了定义这些构造函数的语义，我们使用分级隶属函数m，它对于解释和概念中的每个个体产生区间中的数字$[0,1]$表达个体在解释中属于概念的程度。阈值概念$C_{\bowtie t}$为了$\bowtie \epsilon \{<,\le ,>,\ge \}$然后收集属于C且度为 ⋈ t的所有个体。逻辑$\tau \mathcal{EL}（米）$延伸$\mathcal{EL}$具有阈值概念，其语义是相对于函数m定义的。为了构建适当的分级隶属函数，我们展示了如何使用概念度量∼（概念之间的包含或等价性的分级概括）来定义分级隶属函数${米}_{～}$。然后我们引入一大类概念度量，称为simi-d，其逻辑$\tau \mathcal{EL}（{米}_{～}）$具有良好的算法特性。基本上，我们在$\tau \mathcal{EL}（{米}_{～}）$在没有 TBox 的情况下是 NP/coNP 完全的，对于非循环 TBox 是 PSpace 完全的，对于一般 TBox 是 ExpTime 完全的。实例问题是个例外，它已经是 PSpace 完整的，没有 TBox 的组合复杂性。虽然上限适用于simi-d的所有元素，但我们只能证明simi-d子类的一些硬度结果。本文大大改进并概括了我们在之前三篇会议论文中展示的结果，并提供了我们所有结果的详细证明。